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Theorem coeq12d 5016
Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
Hypotheses
Ref Expression
coeq12d.1  |-  ( ph  ->  A  =  B )
coeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
coeq12d  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )

Proof of Theorem coeq12d
StepHypRef Expression
1 coeq12d.1 . . 3  |-  ( ph  ->  A  =  B )
21coeq1d 5013 . 2  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  C ) )
3 coeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43coeq2d 5014 . 2  |-  ( ph  ->  ( B  o.  C
)  =  ( B  o.  D ) )
52, 4eqtrd 2475 1  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    o. ccom 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-in 3347  df-ss 3354  df-br 4305  df-opab 4363  df-co 4861
This theorem is referenced by:  xpcoid  5390  dfac12lem1  8324  dfac12r  8327  imasval  14461  cofuval  14804  cofu2nd  14807  cofuval2  14809  cofuass  14811  cofurid  14813  setcco  14963  isdir  15414  evl1fval  17774  znval  17978  znle2  17998  mdetfval  18409  ust0  19806  trust  19816  metustexhalfOLD  20150  metustexhalf  20151  isngp  20200  ngppropd  20235  tngval  20237  tngngp2  20250  imsval  24088  opsqrlem3  25558  hmopidmch  25569  hmopidmpj  25570  pjidmco  25597  dfpjop  25598  zhmnrg  26408  dfrtrcl2  27362  mdetdiaglem  30947  istendo  34416  tendoco2  34424  tendoidcl  34425  tendococl  34428  tendoplcbv  34431  tendopl2  34433  tendoplco2  34435  tendodi1  34440  tendodi2  34441  tendo0co2  34444  tendoicl  34452  erngplus2  34460  erngplus2-rN  34468  cdlemk55u1  34621  cdlemk55u  34622  dvaplusgv  34666  dvhopvadd  34750  dvhlveclem  34765  dvhopaddN  34771  dicvaddcl  34847  dihopelvalcpre  34905
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